RATIONAL PRICING OF INTERNET COMPANIES

REVISITED

September 2000

Eduardo S. Schwartz

Anderson School at UCLA

Mark Moon

Fuller & Thaler Asset Management

2

RATONAL PRICING OF INTERNET COMPANIES REVISITED

1. INTRODUCTION

In Schwartz and Moon (2000) we developed a model for pricing Internet

companies using real options theory and modern capital budgeting techniques. The

novelty of the approach is that uncertainty about the key variables which determine the

value of an Internet company play a central role in the valuation. In particular, we

considered uncertainty in both the revenues and the rates of growth in revenues.

In this article we expand and improve the model in several directions. First, we

introduce a third stochastic variable to the model, variable costs. These are allowed to

follow a mean reverting process with volatility that also mean reverts deterministically.

This feature is important since many Internet companies have not yet been profitable but,

presumably, are expected to be profitable in the future.

Second, we use the fact that the theoretical framework allows us to compute the

“beta” of the stock as means of inferring the risk premium in the model. The estimation

of the market price of risk in the model, an unobservable parameter, was one of the most

challenging aspects of the approach. We can now use the beta of the Internet stock to

infer this parameter.

Third, we explicitly take into account capital expenditures and depreciation when

calculating the net after tax cash flows. This is done by introducing a third deterministic

path-dependant state variable - Property, Plant and Equipment – which increases with

3

capital expenditures and decreases with depreciation. This feature of the model is

important for Internet companies that require large investment in fixed assets.

Fourth, we attempt to improve the bankruptcy condition in the model by allowing

the cash balances to become negative. This allows for future equity and debt financing.

The optimal financing is that which maximizes the value of the firm.

Fifth, we suggest a number of simplifying assumptions that considerably facilitate

the practical implementation of the model. One of these is to set all the speeds of

adjustment in the model equal to one another and derive them from the “half-life” of the

company to becoming a “normal” firm. Another is to have only one market price of risk

in the model by assuming that the growth rates in revenues and variable costs are

orthogonal to the “market” returns.

Sixth, we relate the half-life of the deviations to analysts’ (or the evaluator’s)

expectations about future revenues. This comes from the fact that the speed of

adjustment of the growth rate in revenues is the most critical one for valuation purposes.

The expanded model, then, has six state variables, three of which are stochastic

and three of which are deterministic (although path dependant). The three stochastic

variables are revenues, the growth rate in revenues and variable costs. The three

deterministic variables are the amount of cash available, the loss-carry-forward and the

accumulated Property, Plant and Equipment. We solve the problem by Monte Carlo

simulation which can easily deal with this number of state variables and the complex

path-dependencies of the problem.

In Section 2 we present the model with all the new features described above in its

continuous-time version. Since the model is solved by simulation using an interval of

4

time that can be quite long, such as one quarter or one year, Section 3 discusses in detail a

discrete time approximation. Section 4 provides an illustrative example of the approach

by pricing Exodus Communications stock and Section 5 presents comparative statics with

respect to some of the key parameters. Section 6 concludes.

2. CONTINUOUS-TIME MODEL

Consider an Internet company with instantaneous rate of revenues (or sales) at

time t given by R(t). Assume that the dynamics of these revenues are given by the

stochastic differential equation:

(1) 1)()()()(

dztdtttRtdR

σµ +=

where )(tµ , the drift, is the expected rate of growth in revenues and is assumed to follow

a mean reverting process with a long-term average drift µ . That is, the initial very high

growth rates of the Internet firm are assumed to converge stochastically to a more

reasonable and sustainable rate of growth for the industry to which the firm belongs.

(2) 2)())(()( dztdtttd ηµµκµ +−=

The mean-reversion coefficient (κ) affects the rate at which the growth rate is expected to

converge to its long-term average, and κ)/(2ln can be interpreted as the "half-life" of the

deviations in that any growth rate µ is expected to be halved in this time period.

The unanticipated changes in revenues are also assumed to converge

(deterministically) to a more normal level whereas the unanticipated changes in the

expected growth rate are assumed to converge (also deterministically) to zero.

(3) dtttd ))(()( 1 σσκσ −=

(4) dtttd )()( 2ηκη −=

5

The unanticipated changes in the growth rate of revenues and the unanticipated changes

in its drift may be correlated:

(5) dtdzdz 1221 ρ=

The costs at time t have two components. The first is a variable component,

which is assumed to be proportional to the revenues. The second is a fixed component.

(6) FtRttCost += )()()( γ

The variable costs parameter γ(t) in the cost function is also assumed to be

stochastic reflecting the uncertainty about future potential competitors, market share, and

technological developments. It follows the stochastic differential equation:

(7) 33 )())(()( dztdtttd ϕγγκγ +−=

The mean-reversion coefficient (κ3) describes the rate at which the variable costs are

expected to converge to their long-term average, and 32ln κ)/( can be interpreted as the

"half-life" of the deviations in that any deviation γ is expected to be halved in this time

period. The unanticipated changes in variable costs are also assumed to converge

(deterministically) to a more normal level

(8) dtttd ))(()( 4 ϕϕκϕ −=

We also allow for correlation between unanticipated changes in variable costs and

both revenues and growth rates in revenues:

(9) dtdzdz 1331 ρ= (10) dtdzdz 2332 ρ=

The after tax rate of net income to the firm, Y(t), is then given by

(11) )1))(()()(()( ctDeptCosttRtY τ−−−=

6

The corporate tax rate τc in (11) is only paid if there is no loss-carry-forward (i.e.

if the loss-carry-forward is positive the tax rate is zero) and the net income is positive.

The dynamics of the loss-carry-forward are given by:

(12) dL(t) = - Y(t)dt if L(t) > 0

dL(t) = Max [- Y(t)dt , 0] if L(t) = 0

The accumulated Property, Plant and Equipment at time t, P(t), depends on the

rate of capital expenditures for the period, Capx(t), and the corresponding rate of

depreciation, Dep(t). Planned capital expenditures, CX(t), are known for an initial

period and after that they are assumed to be a fraction CR of revenues. Depreciation is

assumed to be a fraction DR of the accumulated Property, Plant and Equipment.

(13) dttDeptCapxtdP )}()({)( −=

(14) ttfortCXtCapx ≤= )()(

ttfortRCRtCapx >= )(*)(

(15) )(*)( tPDRtDep =

Then, the amount of cash available to the firm, given by X(t), evolves according

to:

(16) dttCapxtDeptYtrXtdX )}()()()({)( −++=

The untaxed interest earned on the cash available is included in the dynamics of the cash

available to make the valuation results insensitive to when the cash flows are distributed

to the security-holders. Since in the risk neutral framework we discount risk adjusted

cash flows at the risk free rate of interest, we also need to accumulate cash flows at the

same risk free rate. For all purposes, except for determining the bankruptcy condition,

7

this is equivalent to discounting the cash flows generated by the firm when they occur as

opposed to at the horizon T.

To avoid having to define a dividend policy in the model, we assume that the cash

flow generated by the firm’s operations remains in the firm and earns the risk free rate of

interest. This accumulated cash will be available for distribution to the shareholders at an

arbitrary long-term horizon T, by which time the firm will have reverted to a “normal”

firm.

The firm is assumed to go bankrupt when its cash available reaches a

predetermined negative amount, X*. The purpose of this is to allow for future financing.

The optimal amount of new financing in the future is the one which maximizes the

current value of the firm, though we recognize that in many practical situations firms go

bankrupt before their value reaches zero. To obtain the optimal amount of new financing

we decrease X* until firm value is maximized.

The objective of the model is to determine the value of the Internet firm at the

current time. According to standard theory this value is obtained by discounting the

expected value of the firm at the horizon under the risk neutral measure (the equivalent

martingale measure) at the risk free rate of interest, which for simplicity is assumed to be

constant1. The value of the firm at the horizon T has two components. First, the cash

balance outstanding and second, the value of the firm as a going concern, the value which

is assumed to be a multiple M of the EBITDA at the horizon T:

(17) ])]}()([)({()0( rTQ eTCostTRMTXEV −−∗+=

1 It would be easy to incorporate stochastic interest rates into our framework.

8

The model has three sources of uncertainty. First, there is uncertainty about the

changes in revenues, second, there is uncertainty about the expected rate of growth in

revenues and third, there is uncertainty about the variable costs. We will assume that only

the first source of uncertainty has a risk premium associated with it and later on we will

relate this risk premium to the “beta” of the stock. Under some simplifying assumptions

(see for example Brennan and Schwartz (1982)), the risk adjusted processes for revenues

can be obtained from the true processes as in:

(18) *1)()]()([

)()(

dztdttttRtdR

σσλµ +−=

where the market price of factor risk λ is constant. This market price of risk is equal to

the correlation of the revenue process and the return on aggregate wealth (proxied by the

market portfolio) times the standard deviation of the return on aggregate wealth (proxied

by the market portfolio). Since the rate of growth in revenues process and the variable

cost process do not have risk premiums attached to them the true and the risk adjusted

processes are the same.

Most analysts and investors are more interested in the price of a share than in the

value of the whole company. To obtain the price of a share we need to examine the

capital structure of the company in detail. We need to know how many shares are

currently outstanding and how many shares are likely to be issued to current employee

stock option holders and convertible bondholders. We also need to know how much of

the cash flows will be available to the shareholders after coupon and principal payment to

the bondholders.

9

To simplify the analysis, we assume that if the firm survives options will be

exercised and convertible bonds will be converted into shares.2 This means that in each

of the paths of the simulation where the company does not go bankrupt, we adjust the

number of shares to reflect the exercise of options and convertibles. To obtain the cash

flows available to shareholders from the cash flows available to all security-holders

(which determine the total value of the firm), we subtract the principal and after-tax

coupon payments on the debt and add the payments by option holders at the exercise of

the options. Since we are assuming that the firm pays no dividends, the exercise of the

options and convertibles occurs at their maturity. If all option holders exercise their

options optimally, the above procedure overvalues the stock by undervaluing the options

and convertibles, since there might be some states of the world where the firm survives

but it is not optimal to exercise the options or convert the convertible.

In addition, it is well known, that for diversification purposes, employee stock

options are frequently exercised before maturity, if they are exercisable, to allow for the

sale of the underlying stock. Also, even if they are in the money, not all the options will

be exercised since many of the employees will leave the firm before their options become

vested. However, over time the firm will likely issue additional employee stock options

to existing and new employees. If the number of shares to be issued at exercise and

conversion is small relative to the total number of shares outstanding, the impact of these

issues on the share value is likely to be very small.

Implicit in the model described above is that the value of the firm and the value of

the stock at any point in time are functions of the value of the state variables (revenues,

expected growth in revenues, variable costs, loss-carry-forward, cash balances and

2 The Longstaff and Schwartz (2000) approach could be used to obtain a more precise exercise strategy.

10

accumulated Property, Plant and Equipment) and time. That is, the value of the stock can

be written as:

(18) ),,,,,,( tPXLRSS γµ≡

Applying Ito’s Lemma to this expression we can obtain the dynamics of the stock value

(19) γµγµγµ

γµ

µγγµγγµµ

γµ

ddSdRdSdRdSdSdS

dRSdtSdPSdXSdLSdSdSdRSdS

RR

RRtPXLR

++++

++++++++=2

212

21

221

The volatility of the stock can be derived directly from this expression

(20) 2313122222

222 222)()()( ηϕρσγρσηρϕησσ µγγµγµ

S

VS

S

SS

S

SS

S

S

S

S

SS

S RRR RRR +++++=

The partial derivatives of the stock value with respect to the level of revenues, to

the expected rate of growth in revenues, and to the variable costs are obtained by

simulation3. Since the volatility of the expected growth rate of revenues (η0) is one of the

most critical parameters in the valuation model and is difficult to estimate, equation (20)

is used in the implementation of the model to imply η0 from the observed volatility of the

stock.

Finally, using expression (19) and the continuous time return on the market

portfolio, and noting that only the revenue process has a risk premium associated with it,

we can write the “beta” of the stock as:

(21) 222

)()(

M

R

M

MRMR

M

SM tS

RStS

RSσ

λσσ

σσρσσ

β ===

Expression (21) establishes the relation between the market price of risk in the model and

the “beta” of the stock, and can be used to infer the value of the market price of risk, λ.

11

3. DISCRETE-TIME APPROXIMATION OF THE MODEL

The model developed in the previous section is path-dependent. The cash

available at any point in time, which determines when bankruptcy is triggered, depends

on the whole history of past cash flows. Similarly, the loss-carry-forward and the

depreciation tax shields, which determine when and how much corporate taxes the firm

has to pay, are also path-dependent. These path-dependencies can easily be taken into

account by using Monte Carlo simulation to solve for the value of the Internet company.

In the implementation of the model we assume that all the mean reversion

coefficients are equal and their unique value is inferred from the expected half-life of

their deviations. To perform the simulation we use the discrete version of the risk-

adjusted processes:4

(22) })(]

2

)()()({[ 1

2

)()(εσ

σσλµ ttt

ttt

etRttR∆+∆−−

=∆+

(23) 2

2

)(2

1)1()()( εη

κµµµ

κκκ t

eetett

ttt

∆−∆−∆− −

+−+=∆+

(24) 3

2

)(2

1)1()()( εϕ

κγγγ

κκκ t

eetett

ttt

∆−∆−∆− −

+−+=∆+

where

(25) )1()( 0tt eet κκ σσσ −− −+=

(26) tet κηη −= 0)(

(27) )1()( 0tt eet κκ ϕϕϕ −− −+=

3 The initial value of the revenues (the rate of growth in revenues and the variable costs) is perturbed toobtain new values of the equity from which these derivatives are computed.

12

Equations (25) - (27) are obtained by integrating (3), (4) and (8) with initial values 0σ ,

0η and 0ϕ . 1ε , 2ε and 3ε are standard correlated normal variates.

Note that as time grows the volatilities in the model converge to:

(28) σσ =∞)(

(29) 0)( =∞η

(30) ϕϕ =∞)(

and the growth rate in revenues converges to

(31) µµ =∞)(

This implies that the revenue process converges to

(32) 1)(

)(dzdt

RdR

σµ +=∞∞

The net income after tax is still given by equation (11) where all variables are

measured over the period t∆ . Similarly, the discrete versions of the dynamics of the

loss-carry-forward, the accumulated Property, Plant and Equipment, and the amount of

cash available are immediate

Depending on the situation, the unit of time t∆ chosen for the analysis is one quarter

or one year. The advantage of using the latter is that it allows for smoothing of seasonal

effects and is more consistent with typically annual longer-term analyst projections.

4. ILLUSTRATIVE EXAMPLE

In this section we apply the model described in previous sections to value Exodus

Communications. Exodus is one of the leading providers of Internet web-page hosting

4 Note that there is a typo in the equation corresponding to (23) in Schwartz and Moon (2000).

13

for companies with high-traffic web sites. It was founded in 1994 and by August 2000 it

had a market capitalization of over 22 billion dollars.

The results obtained from our valuation approach, like those of any other valuation

approach, depend critically on the assumptions we make about future revenues, rates of

growth in revenues, and costs. In what follows we describe how we estimate the

parameters needed to run the Monte Carlo simulation from the past and present data

available for Exodus, and from analysts forecasts relating to the company. Since most

forecasts of capital expenditures, revenues and costs are available on an annual basis, all

our simulations are done on an annual basis.

Revenue Dynamics

As the starting value for the revenue simulation we take the actual revenues for

1999 of $242.2 million. The initial volatility of revenues is obtained from quarterly data

from the first quarter of 1997 to the second quarter of 2000, and annualized to give 0.135.

This volatility is assumed to decrease with time to the long-term volatility of revenues of

0.065. The half-life of this deviation and of all the others in the model are assumed to be 2

years. Later on we explain in more detail this last assumption.

Growth Rate of Revenues Dynamics

The initial growth rate in revenues is taken to be 120% per year which is what

analysts expect it to be from 1999 to 2000. It is assumed that in the long run, when

Exodus becomes a “normal” firm, this growth rate will decrease to 5% per year. The

initial annual volatility of the growth rate, which is one of the critical parameters in the

model, is unobservable. We imply it from the volatility of the stock. Figure 1 shows the

weekly implied volatility of Exodus options from September 25, 1998 to August 4, 2000.

14

The average implied volatility of the stock during this period was 96.2%. Figure 2

illustrates the relation between the model price and volatility of the stock and the

volatility of the growth rate of revenues. As an input to the model we choose the

volatility of the growth rate of revenues, 0.23, which gives a model stock volatility which

approximates the market volatility. Note that the corresponding level of the stock price is

$49.92.

Variable Costs Dynamics

Using actual data for 1998 and 1999, and analyst estimates for 2000 to 2009,

Figure 3 reports the results of regressing cash costs on revenues. From these results we

take 0.56 as the initial variable cost as a fraction of revenues, and $382 million as the

annual fixed cost. Since it is unlikely that the firm will continue to have such large profit

margins forever, we assume that in the long run variable costs will increase to $0.75 per

dollar of revenue. Similarly, we assume that the initial volatility of variable costs of 0.06

will decrease in the long run to 0.03.

Half-Life of Deviations and Correlations

As mentioned earlier, to simplify the analysis we have assumed that all the mean

reversion processes in the model have the same speed of adjustment coefficient. But only

the one which determines how fast the initial growth rate in revenues reverts to the long-

term rate has a significant effect on valuation. This is not surprising. If we start with an

initial growth rate of 120% and we assume that in the long run it decreases to a more

normal 5%, then the speed at which it decreases will have a tremendous impact on future

revenues. Figure 4 shows the expected revenues up to year 10 (year 2009) for different

assumptions about the half-life parameter. Note that if we increase the half-life from the

5 The valuation results are not sensitive to this parameter.

15

assumed 2 years to 2.2 years, the expected revenues in year 2009 increases from $18.2

billion to $23.8 billion. Clearly, this should have a major impact on valuation. Finally

note that the analysts’ expectation of eventual revenues of $30.3 billion seems too large

in our view.

In this illustration we have assumed that the three stochastic processes are

uncorrelated. In the next section we look at the effect of possible correlations between

the variables.

Balance Sheet Data

The cash and marketable securities available after two bond issues on July 6, 2000

was $1,720.2 million and the loss-carry-forward at the end of the second quarter of 2000

was $337.9 million. The accumulated property, plan and equipment at the end of 1999

was $390.6 million.

The number of shares outstanding after the stock split of June 20, 2000 was

412.36 million shares. In addition the company had 110.3 million employee stock

options outstanding with a weighted average life of 8.9 years and a weighted average

exercise price of $7.71.6 Finally, the firm had six debt issues and two convertible bond

issues for a total outstanding amount of $2,547 million. This information, together with

the maturity, coupon and conversion ratios (for the convertibles) is used to determine the

price of the stock starting from the total value of the firm.

Capital Expenditures and Depreciation

Analysts have made forecasts of the capital expenditures of Exodus for the next 9

years starting form $1,350 million in year 2000 and decreasing to $600 million in year

2008. We use these as inputs to the model and starting from 2009 we assume that capital

16

expenditures will be 2% of revenues. In accordance with the firm practice we assume

that the annual depreciation allowance is 25% of the accumulated property, plant and

equipment of the previous year.

Environmental and Risk Parameters

The corporate tax rate is taken to be 35% and the risk free rate of interest 6%.

The annual volatility of the market portfolio used in the model to relate the market price

of risk to the beta of the stock is assumed to be 15%.

The market price of risk in the model corresponding to the revenue process is an

unobservable parameter. We imply it from the beta of the stock. Figure 5 shows the

regression used to estimate beta using weekly stock returns for Exodus and the S&P500

from March 27, 1998 to August 4, 2000. The estimated beta is 2.78. Figure 6 reports the

relation between the market price of risk parameter and the stock price and beta. A

market price of risk of 0.37 gives a beta for the stock of 2.74, which is very close to the

estimated beta for Exodus. For this market price of risk the corresponding stock price is

once again $49.92.7

Given its substantial capital expenditure plans, Exodus has future financing plans

of the order of $2.5 billion. Therefore, we assume that Exodus has future financing

possibilities of $3 billion. Increasing or decreasing this amount by $500 million has little

effect on the valuation.

Simulation Parameters

6 The computer program actually uses the more detailed employee stock options data.7 This is no coincidence, since the volatility of the revenue growth and the market price of risk are impliedsimultaneously to give the stock volatility and beta.

17

In all cases we use 10,000 simulations with steps of one year and up to a horizon

of 20 years. At the horizon we assume that the value of the firm is a multiple of ten times

earnings before interest, taxes, depreciation and amortization (EBITDA).

Simulation Results

Using the parameters described above, the model stock price for Exodus

Communications that we obtain is $49.92, with model volatility and beta that closely

match (by construction) observed volatility and beta of the stock. The model (risk

neutral) probability of bankruptcy is 3.4% with default occurring in years 5 to 9 (0.2% in

year 5, 0.8% in year 6, 1.2% in year 7, 0.7% in year 8, and 0.6% in year 9).

On August 9, 2000 the market price of Exodus at the close was $54.75. This is

approximately 10% above the model price.

Figures 7, 8 and 9 show the frequency distribution of revenues, growth rates in

revenues and variable costs, respectively, at the end of year 3 implied by the above

parameters. As can be seen from the figures there is substantial uncertainty about the

state variables of the model even three years in the future. (Note that with only 10,000

simulations the frequency distributions do not appear as smooth as they really are.)

5. SENSITIVITY ANALYSIS

In this section we perform some sensitivity analysis on the more critical or

controversial parameters of the model.

Correlations

Consider the situation in which the firm is in a competitive environment such that

increases in profit margins (i.e. decreases in variable costs) are associated with decreases

in growth rates in revenues. This implies a positive correlation between variable costs

18

and growth rates in revenues. We run the program with the same data described above,

but assuming a correlation of 0.8. The market price of risk and the volatility of the

growth rate had to be adjusted slightly to match the volatility and the beta of the stock.

The model price decreased only slightly to $49.60, but the (risk neutral) probability of

default decreased substantially to 1.9%. The effect of this correlation on prices increases,

however, with the volatility of variable costs.

Variable Costs

As expected long-term variable costs have a big effect on prices. If long-term

variable costs are increased from 0.75 to 0.80, the model stock price goes to $38.91 and if

they are decreased to 0.70, the model stock price goes to $60.35. In both cases small

adjustments are needed in the market price of risk to match the beta of the stock. This

result is not surprising: a 20% increase or decrease in the profit margin has a similar

effect on model stock prices.

Bankruptcy Level

Changes in the amount of future financing allowed has a very small effect on

stock prices since we seem to be close to the optimal amount of future financing. If we

increase the financing constraint from $3 billion to $3.5 billion the stock price increases

only to $50.00 and if we decrease it to $2.5 billion the stock price decreases only to

$49.82. These differences are well within simulation error.

Half-Life of Deviations

As mentioned earlier and as Figure 4 shows, the speed of adjustment has an

important impact on expected future revenues and therefore on valuation. If we increase

the half-life parameter from 2 to 2.2 years, the stock price increases to $73.55 (recall that

19

expected revenues in year 10 increase from $18.2 billion to $23.8 billion). If we decrease

the parameter to 1.8 years the stock price decreases to $33.39 (the expected revenues in

year 10 decrease to $13.9 billion).

6. CONCLUSION

In this article we have substantially extended and perfected the model developed

in Schwartz and Moon (2000) for pricing Internet companies. We introduce uncertainty

in costs and we take into account the tax effects of depreciation. We use the beta and the

volatility of the stock to infer two critical unobservable parameters of the model, the

market price of risk and the volatility of growth rates in revenues. We improve the

bankruptcy condition by allowing future financing by the firm. Finally, we facilitate the

implementation of the model by assuming the half-life of all processes is the same and

suggest that it can be estimated from evaluator’s expectations of future revenues.

Like any other valuation method, our real options approach to value Internet

companies depends critically on the parameters used in the estimation. The next step in

the implementation of the model would be to use cross-sectional data of many Internet

companies to estimate some of the parameters. Finally, since the growth rate in revenues

is not observable “learning models” could be used to update this critical factor in the

model when new information on revenues is revealed.

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References

Brennan, M.J. and E.S. Schwartz, (1982), "Consistent Regulatory Policy Under

Uncertainty," The Bell Journal of Economics, 13,, 2, 507-521 (Autumn).

Longstaff, F.A. and E.S. Schwartz, (2000), “Valuing American Options by Simulation: A

Simple Least-Square Approach”, Review of Financial Studies (forthcoming).

Schwartz, E.S. and M. Moon, (2000), “Rational Pricing of Internet Companies”,

Financial Analysts Journal 56:3, 62-75 (May/June).

21

Figure 1Exodus: Implied Volatility (%)

0

50

100

150

200

250

24-Jul-98 01-Nov-98 09-Feb-99 20-May-99 28-Aug-99 06-Dec-99 15-Mar-00 23-Jun-00 01-Oct-00

Imp

lied

Vo

lati

lity

22

Figure 2Exodus Communications: Valuation

40

42

44

46

48

50

52

54

56

58

0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28

Volatiltiy of the Revenue Growth Rate

Sto

ck P

rice

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Vo

lati

lity

Stock Price

Volatility

0.97

$49.92

23

Figure 3EXDS: Cash Costs vs. Revenues

Not Including CAPX1998A, 1999A, E[2000-2009]

y = 0.5623x + 382.4

R2 = 0.9991

0

2,000

4,000

6,000

8,000

10,000

12,000

14,000

16,000

18,000

20,000

0 5,000 10,000 15,000 20,000 25,000 30,000 35,000

Annual Revenues ($MM)

An

nu

al C

ost

s in

EB

ITD

A (

$MM

)

24

Figure 4Estimated Revenues

0

5000

10000

15000

20000

25000

30000

35000

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009

Year

Rev

enue

Analyst

H-L=2.0

H-L=2.2

H-L=1.8

25

Figure 5EXDS returns vs. S&P500 returns

(weekly data)

y = 2.7758x + 0.031

R2 = 0.2686

-0.500

-0.400

-0.300

-0.200

-0.100

0.000

0.100

0.200

0.300

0.400

0.500

-0.150 -0.100 -0.050 0.000 0.050 0.100 0.150

S&P500 Return

EX

DS

Ret

urn b

26

Figure 6Exodus Communication: Valuation

46

47

48

49

50

51

52

53

54

0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42

Market Price of Risk

Sto

ck P

rice

2

2.2

2.4

2.6

2.8

3

3.2

Bet

a Stock Price

Beta2.74

$49.92

27

Figure 7Frequency of Revenues in Year 3

0

100

200

300

400

500

0 2000 4000 6000 8000 10000

Revenues

Fre

quen

cy

Mean=3300

28

Figure 8Frequency of Growth Rates in Revenues in Year 3

050

100

150

200

250300

350

400

0 0.2 0.4 0.6 0.8 1

Growth Rate in Revenues

Fre

qu

ency

Mean=0.46

29

Figure 9Frequency of Variable Costs in Year 3

0

50

100

150

200

250

300

350

0.5 0.6 0.7 0.8 0.9

Variable Costs

Fre

qu

ency Mean=0.68